Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3541 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,19,22,18,24,17,23)(15,25,27,20,29,38,28,34)(16,21,32,35,31,36,30,37)(26,39,40,33,41,44,42,43);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,26)(17,27)(18,29)(19,28)(21,33)(22,25)(23,38)(24,34)(30,40)(31,41)(32,42)(35,39)(36,43)(37,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,30)(18,31)(19,32)(20,33)(22,35)(23,37)(24,36)(25,39)(27,40)(28,42)(29,41)(34,43)(38,44);(7,12)(9,11)(13,17,18,19)(14,22,24,23)(15,27,29,28)(16,30,31,32)(20,25,34,38)(21,35,36,37)(26,40,41,42)(33,39,43,44);(6,10)(9,11)(16,31)(21,36)(26,41)(30,32)(33,43)(35,37)(39,44)(40,42)Polycyclic presentation:
2^9|a^2=fi,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407319);